Constraint-based probabilistic learning of metabolic pathways from tomato volatiles.

Abstract

Clustering and correlation analysis techniques have become popular tools for the analysis of data produced by metabolomics experiments. The results obtained from these approaches provide an overview of the interactions between objects of interest. Often in these experiments, one is more interested in information about the nature of these relationships, e.g., cause-effect relationships, than in the actual strength of the interactions. Finding such relationships is of crucial importance as most biological processes can only be understood in this way. Bayesian networks allow representation of these cause-effect relationships among variables of interest in terms of whether and how they influence each other given that a third, possibly empty, group of variables is known. This technique also allows the incorporation of prior knowledge as established from the literature or from biologists. The representation as a directed graph of these relationship is highly intuitive and helps to understand these processes. This paper describes how constraint-based Bayesian networks can be applied to metabolomics data and can be used to uncover the important pathways which play a significant role in the ripening of fresh tomatoes. We also show here how this methods of reconstructing pathways is intuitive and performs better than classical techniques. Methods for learning Bayesian network models are powerful tools for the analysis of data of the magnitude as generated by metabolomics experiments. It allows one to model cause-effect relationships and helps in understanding the underlying processes. ELECTRONIC SUPPLEMENTARY MATERIAL: The online version of this article (doi:10.1007/s11306-009-0166-2) contains supplementary material, which is available to authorized users.

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ORIGINAL ARTICLE
Constraint-based probabilistic learning of metabolic pathways
from tomato volatiles
Anand K. Gavai · Yury Tikunov · Remco Ursem ·
Arnaud Bovy · Fred van Eeuwijk · Harm Nijveen ·
Peter J. F. Lucas · Jack A. M. Leunissen
Received: 23 January 2009/Accepted: 28 April 2009/Published online: 30 May 2009
© The Author(s) 2009. This article is published with open access at Springerlink.com
Abstract
have become popular tools for the analysis of data produced
by metabolomics experiments. The results obtained from
these approaches provide an overview of the interactions
between objects of interest. Often in these experiments, one
is more interested in information about the nature of these
relationships, e.g., cause-effect relationships, than in the
actual strength of the interactions. Finding such relation-
ships is of crucial importance as most biological processes
Clustering and correlation analysis techniques
can only be understood in this way. Bayesian networks
allow representation of these cause-effect relationships
among variables of interest in terms of whether and how
they influence each other given that a third, possibly empty,
group of variables is known. This technique also allows the
incorporation of prior knowledge as established from the
literature or from biologists. The representation as a direc-
ted graph of these relationship is highly intuitive and helps
to understand these processes. This paper describes how
constraint-based Bayesian networks can be applied to
metabolomics data and can be used to uncover the impor-
tant pathways which play a significant role in the ripening of
fresh tomatoes. We also show here how this methods of
reconstructing pathways is intuitive and performs better
than classical techniques. Methods for learning Bayesian
network models are powerful tools for the analysis of data
of the magnitude as generated by metabolomics experi-
ments. It allows one to model cause-effect relationships and
helps in understanding the underlying processes.
Keywords
networks · Metabolic pathways · Tomato volatiles ·
oxylipin pathway · urea/citric acid cycles
Constraint-based learning · Bayesian
1 Introduction
Metabolomics plays an increasingly important role in the
research area of drug discovery, food & nutrition, plant and
animal biology and many other applications. Where tran-
scriptomic and proteomic analysis does not tell the
complete story, metabolic profiling can add significantly to
the picture of what is happening inside a living cell. Sta-
tistical and mathematical techniques are commonly used to
correlate changes in metabolic composition with changes in
Electronic supplementary material
this article (doi:10.1007/s11306-009-0166-2) contains
supplementary material, which is available to authorized
users.
The online version of
A. K. Gavai · H. Nijveen · J. A. M. Leunissen (&)
Laboratory of Bioinformatics, Wageningen University,
Wageningen, The Netherlands
e-mail: jack.leunissen@wur.nl
A. K. Gavai
Nutrigenomics Consortium, Top Institute (TI)
Food and Nutrition, Wageningen, The Netherlands
A. K. Gavai
Nutrition, Metabolism and Genomics group, Division of Human
Nutrition, Wageningen University, Wageningen, The
Netherlands
Y. Tikunov · A. Bovy
Plant Research International, Wageningen University,
Wageningen, The Netherlands
R. Ursem · F. van Eeuwijk
Biometris, Wageningen University, Wageningen,
The Netherlands
P. J. F. Lucas
Institute for Computing and Information Sciences,
Radboud University Nijmegen, Nijmegen, The Netherlands
123
Metabolomics (2009) 5:419–428
DOI 10.1007/s11306-009-0166-2

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biological conditions (Suizdak 2003; Eiceman and Karpas
2005; Gohlke 1959; Weckwerth 2003; Kopka et al. 2004).
Chromatography coupled to mass spectrometry based
methods, e.g., GC-MS and LC-MS, have been the most
popular metabolic profiling techniques over the past decade.
Hundreds of new metabolites have been identified in plants
(Fiehn et al. 2000; Moco et al. 2006; Schauer et al. 2006)
and the improved sensitivity of modern methods has led to
an increased amount of metabolic information. Techniques
such as these have enabled identification of metabolites at
much higher resolutions than previously possible. Interest-
ing relationships can thus be found by integrating different
types of data (omics) from various analytical sources. In this
paper, Bayesian network learning methods are explored to
uncover molecular pathways of tomato metabolism. This is
done by using constraint-based learning methods. The
related research is reviewed in the next section.
1.1 Related research
Various methods in classical multivariate statistics have
been used in the past to discover and visualize complex
metabolic networks using supervised and unsupervised
clustering methods. Clustering techniques (Opgen-Rhein
and Strimmer 2007) provide good summarization of data
concerning functional relations between metabolites but as
these methods are global one cannot expect to find relations
which are relevant for small subsets of data. These tech-
niques are good for capturing whether or not variables
influence each other; however the nature of interaction
between metabolites is complex and cannot be estimated
using only linear correlations (Husmeier et al. 2005). Fur-
thermore, domain knowledge, which often plays a vital role
to find novel relationships and which can be obtained from
literature and experts, cannot be incorporated in these tra-
ditional techniques with the exception of choosing the
properparametricformofthefunctionalinteractionbetween
variables, e.g., linear or exponential. Learning logistic
regression models from data is the standard approach for
capturing the statistical interactions among a set of input
variables to predict the value of a dependent, or output,
variable. However, it is difficult to establish the impact of
process changes among the variables using only regression
models. Moreover, when there are insufficient data it cannot
accommodate background knowledge (expert judgement)
and causal explanation to the relationships obtained.
We present here a constraint-based Bayesian network
approach, which is a specialized form of graphical models
(Jordan 2004). Use of Bayesian networks to analyze bio-
logical datasets in various genomics domain has been
growing in the last decade. Different types of Bayesian
network learning methods, e.g., search and score, have
been used to recover target-regulator pairs from a yeast cell
cycle microarray datasets (Murphy 2002); Zou and Conzen
2005), and also significant work has been done by Fried-
man et al. to reconstruct gene regulatory networks from
microarray datasets (Friedman et al. 2000). Similar tech-
niques have also been used for pathway identification to
understand the underlying biological processes. In this
paper we demonstrate how Bayesian network learning
techniques can be used to uncover a very important met-
abolic pathway (oxylipin pathway) which plays an
important role in the ripening of fresh tomatoes. We also
show why this technique is better than other statistical
techniques used in this domain.
In principle a Bayesian network is a graphical represen-
tation of a multivariate probability distribution and is an
example of a probabilistic network. A probabilistic network
typically consists of nodes connected by edges, where each
node corresponds to a random variable and edges represent
dependencebetweenthem.Absenceofedgesbetweennodes
represents conditional independence. Bayesian networks, a
special type of a probabilistic network, contain directed
edges, also called as arcs or arrows. The advantage of
Bayesian networks over alternative techniques (e.g., logistic
regression) is that they allow explicit representation of the
mutual interactions among variables and groups of vari-
ables. They take into account the explanatory power of
known variables in an intuitively simple graphical format.
As a Bayesian network is a multivariate probability distri-
bution with statistical independence assumptions, it is
possible to reason probabilistically with this representation
(For an example see the next section). The other advantage
liesinthefactthatprobabilitydistributionscanbeupdatedin
the light of new, known information. This is why Bayesian
networks can be used to support decision making. The
results obtained using this technique are exceptionally
intuitive and this type of analysis is not possible by classical
analysis tools. Nonetheless, there are a few limitations; for
example when the number of variables increases in size the
computational complexity increases, which is NP-hard.
Apart from this, Bayesian networks, like regression models,
are also sensitive to sample size. However, on the positive
side most biological processes are hierarchical in nature and
there are more variables then relationships between them
i.e., the graphs are sparse.
1.2 Example
In Fig. 1 we present an example of the urea and citric acid
cycles. These two cycles are linked by the synthesis of
fumarate. To highlight the basics of the Bayesian network
method, the reactions leading to and from fumarate are
shown in Fig. 2, where both the probability distribution and
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the graph are shown. The graph encodes rather subtle
information about statistical dependence and independence
between sets of variables. For example, according to the
graph in Fig. 2, the concentrations of both argininosucci-
nate and succinate are independent as there are no arrows
connecting these nodes. Both production of fumarate and
malate are a common consequence of presence of argini-
nosuccinate and succinate (as there are directed paths
going from these nodes to both fumarate and malate). The
semantics attached to Bayesian networks implies that if
either fumarate or malate or oxaloacetate are observed in
high levels, then argininosuccinate and succinate become
dependent given the fact that we know levels of fumarate.
Finally, argininosuccinate (or succinate) and oxaloacetate
are conditionally independent given the concentration
levels of fumarate. It also means that if one has observed
high or low levels of fumarate then this does not convey
any new information about concentrations of malate or
oxaloacetate, and vice versa. Given a Bayesian network,
any conditional probability involving any of the variables
included in the model can be computed.
It is a standard practice to compute probabilities of
individual variables from a set of variables. These proba-
bilities are referred as marginal or conditional and are
updated by fixing observations over one or more variables.
Figure 3 shows the bar graphs associated with the
individual variables when the frequency of states of these
variables are computed; in Fig. 4 the variables have been
conditioned on the assumption that oxaloacetate=high to
display that for the case when the concentration is higher
Fig. 1
Urea/Citric Acid cycle
Fig. 2
probability distribution Pr and a directed graph. The probability
distribution Pr is specified using conditional probability distribution
associated to the individual nodes, such as Pr(A = high · AS = high)
= 0.96
Example of a simple Bayesian network consisting of a
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for a certain metabolite. In both figures simply looking at
the shape of the bar graphs already conveys much infor-
mation on the concentration levels of each metabolite.
1.3 Overview of Bayesian networks
Bayesian networks can be learned from data as a standard
practice in multivariate statistics, but as they are easily
understood they can also be manually constructed based on
expert knowledge in a particular problem domain. For
example, if one knows the interaction of metabolites in a
certain pathway, one can even make a hypothetical network
based on literature without data. Each of the metabolites in
the network would be associated with a probability
embedded in a contingency table (also known as a condi-
tional probability table), expressing an expert’s degree of
belief. As was illustrated above, if one has observed a level
of one or more compounds it is possible, using the Bayesian
network, to predict the likelihood or concentration levels of
the other compounds. Thus the likelihood represents the
concentration levels of metabolites. The important aspects
of understanding Bayesian networks lies in the fact that the
graph structure of network is separate from the probability
distribution associated with it.
The graphical nature of the network combined with
probability theory allows one to do data analysis in an intu-
itive way. It is important to understand the interaction
between different variables but it is more important to
understandthenatureoftheserelationships.Fromexamplein
the previous section representing urea/citric acid cycle in
Fig. 1 argininosuccinate, fumarate and malate represent a
serial connection, arginine, argininosuccinate and fumarate
represent a diverging connection and argininosuccinate,
fumarate and succinate represent a converging connection.
As mentioned before knowing information about the con-
centrations levels of fumarate makes argininosuccinate and
malate independent in a serial connection, knowing con-
centration levels of argininosuccinate make arginine and
fumarateindependentinadivergingconnectionandknowing
concentration levels of fumarate makes argininosuccinate
and succinate dependent in a converging connection.
Formally, a Bayesian network represents an acyclic
directed graph (ADG), i.e., a set of nodes, or vertices, and
directed edges, or arcs, and is defined as a pair G = (V,E),
where V is a finite set of distinct nodes and E ? V ? V is a
set of distinct arcs. A pair ðu;vÞ 2 E is denoted by an arc
u → v from u to v and it is said that u is a parent of v and v is
saidtobeachildofu,oftenthisrelationshipalsorepresentsa
Fig. 3
probability distributions for the
Bayesian belief network shown
in Fig. 2
Prior marginal
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cause-effect relationship. A path hv1;...;vni is a set of dis-
tinctnodessuchthatvi→vi+1orvi←vi+1,foreachi.Apath
is called directed if vi← vi+1or vi→ vi+1, for each i. A node
v is called a descendant of a node u if there is a directed path
from u to v in the graph. Statements on conditional depen-
denceandindependencecanbederivedfromthegraphusing
the d-separation criterion (Pearl 1998). The basic idea of d-
seperation is that just by looking at the graph it is possible to
derive independence information about the associated
probability distribution of a Bayesian network. Consider
Fig. 5, which depicts three types of subgraphs which are
relevant in interpreting a Bayesian network. The subgraphs
X → Y → Z and X ← Y ← Z, called serial connection and
X ← Y → Z, called diverging connection, are equivalent.
Then there is also the non-equivalent converging connec-
tion. All three of these causal situations give rise to different
conditional independence of the associated random. For-
mally d-separationisexpressed asfollows: Twonodes onan
ADG, G = (V,E) are said to be d-separated by a set of nodes
S ? V if there is a node w such that either:
●
w 2 S and w does not have a converging connection on
any path connecting the nodes and w is on this path, or
●
w 62 S and neither any of the descendants of w in S.
If two variables are d-separated given a set of variables
S in a directed graph, then they are conditionally inde-
pendent given another set of variables in all probability
distributions compatible with the graph. Two variables X
and Y are conditionally independent given a set of variables
Fig. 4 Posterior marginal
probability distributions for the
Bayesian belief network after
entering evidence on
concentration levels of
oxaloacetate. Note the increase
in probabilities of the levels of
concentrations of both
oxaloacetate and
argininosuccinate compared to
Fig. 3. It also predicts that it is
more likely that the
%concentration levels of
argininosuccinate to be high
a
b
c
Fig. 5
their interpretation
Three different graph structures of a Bayesian network and
Constraint-based probabilistic learning of metabolic pathways423
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S if knowledge on X gives no additional information on Y
once we know about S.
Bayesian networks are also said to be an independence
map, or I-map for short, as independencies can be read off
by d-separation from the graph and these also hold for the
underlying probability distribution. This property allows us
to find independence between variables of interest in a
problem domain. d-Separation in a graph structure G is
represented by ⫫G and conditional independence in a
probability distribution, by ⫫P. Some standard notations
used in the context are as follows:
●
u ⫫Gv∣S : u 2 V and v 2 V are d-separated in graph
G = (V,E), given a set of nodes W.
U ⫫GV∣W : Each u 2 V and each v 2 V is d-separated
in graph G or U and V are d-separated in G, given the set
of nodes W.
U
GV∣W : Each u 2 V and each v 2 V is d-connected,
given the set of nodes W.
X ⫫PY∣S : X and Y are conditionally independent
given S.
X
PY∣S : X and Y are conditionally dependent given
S.
if S = /, and X ⫫PY ∣ / holds, then sets X and Y are
called marginally independent.
●
●
●
●
●
Not always we will have knowledge about relationships
between metabolites of interest in which case we would
want to find these relationships from available experimental
data. This approach is not uncommon when the number of
variablesislargeandthereislittleornoknowledgeavailable
of the underlying process. Moreover, it can be a laborious
task to construct networks of several hundred nodes just by
hand. Therefore there has been considerable research in this
area to do unsupervised learning of conditional dependence
and independence relationships from data. The key
assumptions used for this approach are that biological pro-
cesses are hierarchical in nature and links between
metabolites in metabolic processes are sparse in nature. In
the past, correlation and clustering methods have been used
successfully to identify groups of metabolites clustering
together and reconstruct pathways (Yilmaz 2001; Ursem
et al. (2008)). Bayesian networks allows us to model causal
relationships by looking at the direction of the arrows in the
directed graph. Causal relationships play a vital role when
we want to find out when one variable causes a change in
another variable. Often these relations are investigated by
experimental research to determine if changes in one vari-
able truly cause change in another variable. However to
generate an equivalent network is still possible using
Bayesian network. For Bayesian networks there is the
restriction that arrows are not allowed to form directed
cycles (paths that end at the node where they started)–these
graphs are called acyclic. Of course, there can be feedback
loops involved in a problem domain which can be perfectly
modeled using another type of Bayesian networks, so-called
dynamic Bayesian networks (Murphy 2002), which require
time-series data.
1.4 Learning in Bayesian networks
Learning the graph of a Bayesian network is done by
exploring data using partial correlations as a means to
distinguish dependent from independent relationships. To
put it simple, direct and indirect relationships can be
identified easily from the constructed networks: metabo-
lites missing arrows indicate (conditional) independence.
There are two aspects of representing data using this
technique viz. qualitative and quantitative. The qualitative
aspect includes representation of data using nodes and
arrows and these relationships can be quantified using a
conditional probabilitydistribution.
methods have the advantage that they allow incorporation
of prior biological knowledge about dependence of vari-
ables, and therefore this has been taken as the method of
choice for the present research. We consider here the PC-
algorithm (Peter and Clark) (Sprites et al. 2000) which is a
constraint-based method. There are certain assumptions to
this approach such as the independence between nodes has
a perfect representation by an ADG; under this assumption
the PC algorithm will discover an equivalent Bayesian
network. Another assumption is that networks are sparse,
i.e., have few relationships between metabolites as shown
in Fig. 1. There are several ways to verify conditional
independence relationships which include reducing size of
the database, finding correlations, direct query from experts
and finding clusters in a causal network.
The algorithm is based on asking true independence
relationship between sets of variables of the form Xi⫫Xj∣S,
where S is a subset of variables. An overview of the steps
are as follows:
Constraint-based
●
●
Construct an undirected graph.
Find converging connections, by testing for indepen-
dence and
Give directions to the links without producing cycles.
●
Here we consider an imaginary oracle as our expert which
tells us if two nodes are conditionally independent given a
subset of nodes S (later this oracle will be replaced by
statistical test to find partial correlations, e.g., G2test or
Fisher’s Z transformation). If the oracle (domain expert)
says two nodes are conditionally independent given a third
node S then we remove the edge between two nodes and
make them independent. Asking questions like this for all
the nodes involved and recursively deleting edges between
nodes based on the answers will result in an undirected
graph also known as the skeleton of the network. The next
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step then is to give directions to the edges based on rules
(Meek 1995b) to generate a ADG, sometime there might
exist bi-directional arrows which is not a part of Bayesian
networks, but its presence indicate hidden nodes, which
might have been missed from the experiment or not being
observed.
In this paper, we demonstrate how learning and rea-
soning with Bayesian networks can be used to reconstruct
the oxylipin pathway found in the synthesis of fresh tomato
volatiles. We show how the results obtained by running
Bayesian network analysis on experimental data of bio-
chemical compounds of beef, round and cherry tomatoes
supports the elucidation of the nature of the biological
processes. For brevity, our focus is only on learning of
structures and not on parameter estimation, which is often
used to learn the probability distribution of the dataset.
Subsequently, we discuss how to interpret the graphs
generated by Bayesian network learning (which is an
important aspect of data analysis). Prior knowledge
obtained from literature is also taken into account in the
form of metabolite selection for the analysis. The dataset
used includes data of tomato volatile metabolite profiling,
as described in (Tikunov et al. 2005). A detailed descrip-
tion of this dataset can be found below in the materials
sections of the article.
2 Materials and methods
2.1 Description of the metabolite dataset
Flavors in tomatoes are important targets for plant breeders
to improve the quality of fresh tomatoes. Therefore it has
become a popular area of research among molecular biol-
ogists to study the pathways involved in biosynthesis, of
the oxylipin (lipoxygenase) pathway of volatile compounds
(VOC, volatiles) in tomatoes. In plants the substrates of
these pathways are linoleic and linolenic acid, while their
mammalian equivalents are arachidonic and eicosapenta-
enoic acids. Tomato volatiles are generally divided into six
groups (Yilmaz 2001) lipid derived, carotenoid related,
amino acid related, terpenoids, lignin related and miscel-
laneous. Each group participates in different pathways
involved in the biosynthesis of the aroma volatiles. Figure 6
shows formation of lipid-derived volatiles through one of
these pathways. There has been substantial research in this
area, however the exact nature of the relationship between
volatile compounds involved is still unknown.
Volatiles have been analyzed using gas chromatography
mass spectrometry in ripe fruits of 94 tomato (Solanum
lycopersicum L.) varieties as described in Tikunov et al.
(Tikunov et al. 2005). The varieties selected represent a
considerable collection of genetic and therefore phenotypic
variation. 322 VOC have been detected and 69 VOC
identified most reliably have been chosen for the present
study. This set of 69 VOC contains metabolites of 7 bio-
chemical groups: volatiles derived from lipids, two
phenylalanine derived groups, leucine and/or isoleucine
derived volatiles, open-chain carotenoid derivatives, cyclic
carotenoid derivatives and terpenoids. The last three groups
are biochemically related and called isoprenoids.
2.2 Analysis
We consider here finding relationships between volatile
metabolites of lipid derivatives involved in the oxylipin/
lipoxygenase (LOX) pathway which occurs during ripening
of tomato fruit (Yilmaz 2001) as depicted in Fig. 6. We
used the package pcalg (Kalisch and Buhlmann 2007)
which is an implementation of the PC algorithm in R
(R Development Core Team 2008) which is an open source
environment for statistical computing to perform the
analysis on a real-life dataset as mentioned in the Sect. 3 of
the article. The inputs were the tomato dataset and the
algorithm allows to set a threshold to find significant
conditional (independent) relationships between these
metabolites. The algorithm generates a graph object which
is an ADG. A bidirectional arrow in such network implies
presence of hidden variables from the experiment being
conducted (hidden factors or latent variables), e.g., when a
Fig. 6
the oxylipin pathway
Formation of lipid-derived volatiles through biosynthesis in
Constraint-based probabilistic learning of metabolic pathways425
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metabolite is missing from the experiment (Beal et al.
2005). Here we show how this algorithm performs on a
real-life dataset and finds relationships among metabolites
of interest which are biologically meaningful and by taking
into account the prior knowledge the generated network is
compared to the established knowledge found in literature.
This can be done by counting missing edges (false nega-
tives) and extra edges (false positives) which were
computed for these metabolites by the algorithm. Finally
the structure Hamming distance metric (Tsamardinos et al.
2006) a measure to calculate the number of substitutions
required to transform one graph to another, is used to
calculate the distance (difference) of the computed network
from the actual network (pathway). The lower this score is,
the better the PC algorithm has performed on the dataset.
The visualization of complex networks is not easy and
the knowledge represented by them is sometimes not
obvious just by looking at these complex networks. The key
to solving these issues is to make use of an interactive graph
which allows looking at the chemical structures and getting
relevant information from online repositories using well-
established visualization techniques. We therefore imple-
mented a framework which handles these issues, and the
networks shown in this paper have been created using the
tools. The tool was developed using open source software
and standards, such as Graphviz (http://www.graphviz.org)
andSVG(http://www.w3.org/Graphics/SVG/).
scripts used for construction of networks are available from
the authors upon request.
Allthe
3 Results and discussion
A primary interest in biology is finding novel biochemical
pathways describing relationships between metabolites and
their dependence on environmental factors. Often the data
arising from experiments are not completely observed; this
scenario is represented as “data being incomplete and the
structure of the Bayesian network being unknown”.
Clearly, this is also the most difficult case from a compu-
tational point of view. Incomplete data may involve
missing values, and imputation techniques are employed to
find appropriate values; these values are assumed to be
missing at random. Sometimes particular crucial variables
are missing, called hidden or latent variables. Using opti-
mization algorithms such as the Expectation Maximization
(EM) algorithm, it is sometimes possible to learn all
parameters, including the hidden ones (Dellaert 2002 &
Elidan and Friedman 2003).
In this study we show how parts of a plant metabolic
system can be reconstructed and visualized by applying
Bayesian networks. We focused on 69 volatile compounds
and the choice of these metabolites was based upon prior
knowledge obtained from the domain experts and relevant
literature (Tikunov et al. 2005; Yilmaz 2001; Yilmaz et al.
2001). We used constraint-based learning of Bayesian
networks with a very low significance level α of 0.0001 on
this dataset. The test statistic used to find the relationship
and their strength are based on Fisher’s Z transformation
(Kalisch and Buhlmann 2007). A Bayesian network esti-
mates a ADG and so relationships do not form a cycle,
observing all such relationships indicate hidden variables
which might have been missed by chance or not being
observed in the experiment. As Bayesian networks gener-
ate equivalent structures, considering the example from
Fig. 5, subgraphs A → B → C, A ← B ← C and
A ← B → C are equivalent. Therefore the methods
described in the analysis section estimated 13 arcs; when
comparing this estimated network with the relationships
found in the literature we were able to find 66 % true
positives, 7 % false positives and a structure Hamming
distance of 9, meaning that it would take 9 operations of
adding, deleting and changing the direction of arrows to
reach the true graph. Analysis of the experiment using the
search-and-score (Heckerman 1995) method produced a
graph with 16 arcs, and contained only 25% true positives
and 56% false positives (the corresponding network graph
can be found in the supplementary material) and a structure
Hamming distance of 15. The true positives and the
structure hamming distance are influenced by the fact that
not all metabolites are assigned, absence of metabolites
from the experiment and unknown relationships. Never-
theless, these figures still are useful to measure the
performance of these techniques. Quantitatively analyzing
techniques like these are common practice, but the real
advantage lies in the graphical representation which is
much more intuitive than standard statistical tests. From
Fig. 6 it can be seen that the enzymes involved in these
pathways are generally known to oxidize certain fatty acids
containing a cis, cis-1, 4-pentadine structure. The main
substrate is therefore linoleic acid (C18:2) and linolenic
acid (C18:3) as shown in Fig. 6. The upper part in Fig. 6
which consist of phospholipids, galactolipids and tria-
cylglcerols has not been taken into account in the
experiment in question as these metabolites are large
chemical structures and therefore are not volatile.
In principle the relationships (correlation and causal)
generated, can be compared to the relationships found by
Tikunov et al. (2005). We consider here 13 metabolites
1-pentene-3-ol, 1-penten-3-one, E-2-pentenal, 1-pentanol,
Z-2-penten-1-ol,
Z-3-hexenal,
Z-3-hexenol, 1-hexanol, heptanal, E-2-heptenal, n-pentanal
which are involved in the substrate formation of free fatty
acids (lower section of Fig. 6) of the oxylipin pathway. As
the exact nature of these relationship is not known (Yilmaz
et al. 2001) a plausible explanation is still possible by
hexanal,
E-2-hexenal,
426A. K. Gavai et al.
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Page 9

looking at the chemical structures of these metabolites.
Figure 7 indicates such a network constructed using
Bayesian approach showing metabolites 1-pentene-3-ol,
1-pentene-3-one,
2-hexenal,
E-2-heptanal
showsignificant
E-2-hexenal is derived by isomerization of Z-3-hexenal
(Baldwin et al. 2000) and the relationship of these two
compounds cannot be observed in the graph; the reason for
this could be absence of isomerization factor or less
number of samples. A relationship between 1-penten-3-one
and 1-pentan-3-ol also makes sense since the first is a
dehydrogenation product of the second. From Fig. 7 correct
relationships from 1-pentene 3-ol → 1-pentene-3-one, E-2
pentenal → Z-2-penten-1-ol and 1-pentanol → n-pentanal
can be deduced. There are certain relationships which may
not make sense just by looking at them, e.g., 1-hexa-
nol → Z-3-hexenol, but the advantage is such relationships
could be easily explained using Bayesian networks which
may indicate present of hidden variables (latent variables)
as not all the metabolites were observed in the experiment.
Similarly, bi-directional arrows also indicate presence of
such variables. To deduce exact relationships is difficult
but the advantage lies in searching for equivalent rela-
tionships which can be easily deduced such as Z-3-
hexenal → Z-3-hexenol can also be seen in Fig. 7.
E-2-pentenal,
causal
heptanal,
relationships.
4 Concluding remarks
Constructing graphically intuitive models has become a
populartechniqueinmetabolomicsexperiments(Morgenthal
et al. 2006; Beal et al. 2005). Models like this allow us to
understand the underlying biological processes involved in
metabolic networks and reconstruct pathways. This knowl-
edge is normally visualized by means of a directed graph,
where the nodes of the graph correspond to variables and
arrowsinthegraphareusedtoexpressstatisticaldependence
and independence information.
The approach described in the present study proved
useful to discover causal biochemical relationships in
complex metabolomics data. The results are confirmed by
previous observations on the same data as well as infor-
mation found in literature. Methods such as Bayesian
networks which are used for causal modeling of high-
dimensional data are powerful tools in modeling of com-
plex systems, since these approaches do take into account
correlation methods before constructing an equivalent or
exact causal relationship. Here we show how a Bayesian
network can be used to analyze metabolomics data which is
a powerful technique and helps us to get indepth under-
standing of the biological process. This method can be
exploited further by coupling it with pathway databases in
order to get exact and more plausible information to
understand the process changes at hand. As more and more
data become available these methods can outperform
classical statistical techniques and be used to find novel
biochemical pathways. We have shown here how this
approach can be used for exploratory data analysis in
searching for causal relationships in metabolomics. The
resulting hypothesis can then be used to form the basis of
subsequent analysis which can learn from data, take prior
inputs from molecular biologist and update probabilities in
the light of “new information” and/or “data”.
Acknowledgments
Genomics (CBSG; http://www.cbsg.nl) for making the data available.
This study was supported by grants from Biorange (http://www.nbic.nl)
and Nutrigenomics Consortium (NGC; http://www.nutrigenomics
consortium.nl).
The authors like to thank Centre for BioSystems
Open Access
ative Commons Attribution Noncommercial License which permits
anynoncommercialuse, distribution, and reproduction inany medium,
provided the original author(s) and source are credited.
This article is distributed under the terms of the Cre-
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